Related papers: Set Theory and p-adic Algebras

For a $C_0(X)$-algebra $A$, we study $C(K)$-algebras $B$ that we regard as compactifications of $A$, generalising the notion of (the algebra of continuous functions on) a compactification of a completely regular space. We show that $A$…

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the…

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mapping in Hausdorff p-vector spaces, and the fixed point theorem for upper semicontinuous set-valued mappings in Hausdorff locally…

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it…

For a compactification $\alpha X$ of a Tychonoff space $X$, the algebra of all functions $f\in C(X)$ that are continuously extendable over $% \alpha X$ is denoted by $C_{\alpha}(X)$. It is shown that, in a model of $\textbf{ZF}$, it may…

One point compactification is studied in the light of ideal of subsets of $\mathbb{N}$. $\mathcal{I}$-proper map is introduced and showed that a continuous map can be extended continuously to the one point $\mathcal{I}$-compactification if…

For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X]^{<\omega}\subseteq\mathcal{Z}\subseteq\mathcal{P}(X)$, and any $\mathcal{A}\subseteq\mathcal{P}(X)$, the topology…

We introduce a new metric on the ideal space of an AF algebra that metrizes the Fell topology. The novelty of this metric lies in the use of a Hamming distance type metric in its construction. Furthermore, this metric captures more of the…

In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the locales of which the Stone-Cech compactification can be defined in constructive…

In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this…

The topological reconstruction problem asks how much information about a topological space can be recovered from its point-complement subspaces. If the whole space can be recovered in this way, it is called reconstructible. Our main result…

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity.…

Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is…

Our aim is to study certain algebraic properties of the ring $C(X)_\mathcal{P}$ of real-valued functions on $X$ whose closure of discontinuity set is in an ideal of closed sets. We characterize $\mathcal{P}P$-spaces using $z$-ideals and…

This paper investigates the absolute values on $\mathbb{Z}$ valued in the upper reals (i.e. reals for which only a right Dedekind section is given). These necessarily include multiplicative seminorms corresponding to the finite prime fields…

Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let…

The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are…

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every…

Since their inception perfectoid spaces have catalyzed a revolution in p-adic geometry. We redevelop the foundations of perfectoid spaces from the point of view of Berkovich Spaces, where the underlying topological space of an affinoid…

In this paper, it is shown that a topological space $X$ is compact iff every maximal ideal of the power set ring $\mathcal{P}(X)$ converges to exactly one point of $X$. Then as an application, simple and ring-theoretic proofs are provided…